| No. of papers | 1 | 9 | 32 | 39 | 43 | 22 | 6 | 2 | ||
| Rated at | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
All together there were 154 papers from the eighth grade, so that if they were arranged in order according to their merit we might begin at the poorest and count through 77 of them (n/2 = 154/2 = 77) to find the median point, which would lie between the 77th and the 78th in quality. If we begin with the 1 composition rated at 0 and count up through the 9 rated at 1 and the 32 rated at 2 in the above distribution, we shall have counted 42. In order to count out 77 cases, then, it will be necessary to count out 35 of the 39 cases rated at 3.
Now we know (if the instructions given above have been followed) that the compositions rated at 3 were so rated by virtue of the fact that the judges considered them nearer in quality to the sample valued at 3.69 than to any other sample on the scale. We should expect, then, to find that some of those rated at 3 were only slightly nearer to the sample valued at 3.69 than they were to the sample valued at 2.60, while others were only slightly nearer to 3.69 than they were to 4.74. Just how the 39 compositions rated on 3 were distributed between these two extremes we do not know, but the best single assumption to make is that they are distributed at equal intervals on step 3. Assuming, then, that the papers rated at 3 are distributed evenly over that step, we shall have covered .90 (35/39 = .897 = .90) of the entire step 3 by the time we have counted out 35 of the 39 papers falling on this step.
It now becomes necessary to examine more closely just what are the limits of step 3. It is evident from what has been said above that 3.69 is the middle step 3 and that step 3 extends downward from 3.69 halfway to 2.60, and upward from 3.69 halfway to 4.74. The table given below shows the range and the length of each step in the Hillegas Scale for English Composition.
| Step No. | Value or Sample | Range of Step | Length of Step |
|---|---|---|---|
| 0. . . . | 0 | 0- .91[32] | .91 |
| 1. . . . | 1.83 | .92-2.21 | 1.30 |
| 2. . . . | 2.60 | 2.22-3.14 | .93 |
| 3. . . . | 3.69 | 3.15-4.21 | 1.07 |
| 4. . . . | 4.74 | 4.22-5.29 | 1.08 |
| 5. . . . | 5.85 | 5.30-6.30 | 1.00 |
| 6. . . . | 6.75 | 6.30-7.23 | .93 |
| 7. . . . | 7.72 | 7.24-8.05 | .81 |
| 8. . . . | 8.38 | 8.05-8.87 | .82 |
| 9. . . . | 9.37 | 8.88- |
From the above table we find that step 3 has a length of 1.07 units. If we count out 35 of the 39 papers, or, in other words, if we pass upward into the step .90 of the total distance (1.07 units), we shall arrive at a point .96 units (.90 × 1.07 = .96) above the lower limit of step 3, which we find from the table is 3.15. Adding .96 to 3.15 gives 4.11 as the median point of this eighth grade distribution.
The median and the percentiles of any distribution of scores on the Hillegas scale may be determined in a manner similar to that illustrated above, if the scores are assigned to the individual papers according to the directions outlined above.
A similar method of calculation is employed in discovering the limits within which the middle fifty per cent of the cases fall. It often seems fairer to ask, after the upper twenty-five per cent of the children who would probably do successful work even without very adequate teaching have been eliminated, and the lower twenty-five per cent who are possibly so lacking in capacity that teaching may not be thought to affect them very largely have been left out of consideration, what is the achievement of the middle fifty per cent. To measure this achievement it is necessary to have the whole distribution and to count off twenty-five per cent, counting in from the upper end, and then twenty-five per cent, counting in from the lower end of the distribution. The points found can then be used in a statement in which the limits within which the middle fifty per cent of the cases fall. Using the same figures that are given above for scores in English composition, the lower limit is 2.64 and the limit which marks the point above which the upper twenty-five per cent of the cases are to be found is 5.08. The limits, therefore, within which the middle fifty per cent of the cases fall are from 2.64 to 5.08.
It is desirable to measure the relationship existing between the achievements (or other traits) of groups. In order to express such relationship in a single figure the coefficient or correlation is used. This measure appears frequently in the literature of education and will be briefly explained. The formula for finding the coefficient of correlation can be understood from examples of its application.
Let us suppose a group of seven individuals whose scores in terms of problems solved correctly and of words spelled correctly are as follows:[33]